Problem: Simplify the following expression: $ r = \dfrac{6z + 5}{z + 4} + \dfrac{-8}{9} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{9}{9}$ $ \dfrac{6z + 5}{z + 4} \times \dfrac{9}{9} = \dfrac{54z + 45}{9z + 36} $ Multiply the second expression by $\dfrac{z + 4}{z + 4}$ $ \dfrac{-8}{9} \times \dfrac{z + 4}{z + 4} = \dfrac{-8z - 32}{9z + 36} $ Therefore $ r = \dfrac{54z + 45}{9z + 36} + \dfrac{-8z - 32}{9z + 36} $ Now the expressions have the same denominator we can simply add the numerators: $r = \dfrac{54z + 45 - 8z - 32}{9z + 36} $ $r = \dfrac{46z + 13}{9z + 36}$